What is the shortcut for orientation?
While both "Transformation Orientation" options have the same tool-tip, they are used at different times.

The left one you marked, is used for gizmos and only when you grab the gizmo and use it, overwrites the "globally" used one to the right.

While the right one is used when you use the G,R and S keys to move, rotate, scale respectively as well as other functions like E for extrude etc. 

Only if you use gizmos, will the left one be used, as it also only becomes available when the gizmo is activated. It can be seen as giving the tool it's own specific orientation option.

The documentation is sadly referring to the same link/explanation for both menus.
After looking through the key-mapping, and checking in the console regarding what is reported when changing the selection for either of the two transformation orientation options, i found, that both set the exact same variable:

bpy.context.scene.type = 'LOCAL'

So using , you only get the pie menu to change the right "globally" used "Transformation Orientation", and it seems to me that there is no key for the tool specific options on the left, neither for a pie menu to change them. And adding a shortcut for those tool specific transformation orientations is not possible either.

That sadly means, that there is no solution i can offer you aside the explanation.